Generalized cross-covariances and their estimation

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Mathematical Geology, Vol. 29, No. 6, 1997

Generalized Cross-Covariances and Their

Estimation 1

H. R. Kiinsch, 2 A. Papritz, 3 and F. Bassi 2

Generalized cross-covariances describe the linear relationships between spatial variables observed at different locations. They are invariant under translation o f the locations f o r any intrinsic pro- cesses, they determine the cokriging predictors without additional assumptions and they are unique up to linear functions. I f the model is stationary, that is i f the variograms are bounded, they correspond to the stationary cross-covariances. Under some symmetry condition they are equal to minus the usual cross-variogram. We present a method to estimate these generalized cross-covar- iances from data observed at arbitrary sampling locations. In particular we do not require that all variables are observed at the same points. For fitting a linear coregionalization model we combine this new method with a standard algorithm which ensures positive definite coregionalization ma- trices. We study the behavior o f the method both by computing variances exactly and by simulating from various models.

KEY WORDS: multivariate intrinsic processes, cokriging, cross-correlations, variance of increments, undersampling, coregionalization.

INTRODUCTION

Cokriging is a powerful method which exploits linear relations between several spatial variables for predicting values at unsampled locations or block means. Wackernagel (1995) considers in detail its various theoretical and practical as- pects, but unlike univariate kriging there are unresolved issues in the procedure. One has to decide which function to use for expressing linear relations between variables: the cross-variogram, the pseudo cross-variogram (Myers, 1991), or the cross-covariances. Each option imposes certain restrictions on the models one can consider and on the sampling scheme. To estimate these functions from data we have to assume that they are invariant under translation, that is they depend only on the relative positions of any two locations. For any intrinsic model the cross-variogram has this property, but the cross-covariances are trans- ~Received 9 September 1996; revised 8 December 1996.

2ETH Zurich, Seminar for Statistics, Sonneggstrasse 33, CH-8092 Zurich, Switzerland. e-mail: kuensch@stat.math.ethz.ch and bassi@math.ethz.ch

3 ETH Zurich, Soil Physics, Institute of Terrestrial Ecology, Grabenstr. 3, CH-8952 Schlieren, Switzerland. e-mail: papritz@ito.umnw.ethz.ch

779

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780 Kiinsch, Papritz, and Bassi

lation-invariant only if the processes are stationary. The pseudo cross-variograrn is intermediate: it is shifi-invariant for some nonstationary models, but not for general intrinsic processes (Papritz, Kiinsch, and Webster, 1993). However, the cross-variogram has other disadvantages: it cannot be estimated unless both variables are observed at many sampling locations, and it is sufficient for cokriging only if an additional symmetry condition holds [Condition (iii) of Myers, 1982], which is equivalent to the symmetry of the cross-covariances in the stationary situations. The pseudo cross-variogram or the cross-variances do not have either of these limitations.

In this paper we investigate a different function for expressing linear relations which is termed the generalized cross-covariance. It is free from any of the noted restrictions. It is shift-invariant for any intrinsic model, it can be estimated from arbitrary sampling schemes and it determines the cokriging pre- dictor without additional assumptions. First, we define the new function and discuss its properties, and then, we present a new estimation method for general sampling schemes. The first part is of more theoretical interest because the definition of the new function is not explicit except in symmetric situations where it is equal to minus the standard cross-variogram. Although in most soil science applications the symmetry assumption is plausible, checking it may be helpful. Our results make this possible without assuming stationarity. Moreover the new estimation method is of high practical importance because one variable may be undersampled in many applications. It is the same idea which was presented in Papritz, Kiinsch, and Webster (1993), namely least-squares fitting of the cloud of products of differences to estimate the expected value of these products. But here we show how one can obtain a computationally feasible solution to this minimization problem. The method can be applied to any parametric model for the generalized cross-covariance. In particular, by selecting a piecewise constant function, we obtain the experimental generalized cross- covariance. But this method does not necessarily give an estimate which is conditionally positive definite. In order to guarantee positive definiteness in the coregionalization model, we combine our estimation method with the algorithm of Goulard and Voltz (1992). When one variable is undersampled, the experimental auto- and cross-variograms are estimated for different lag distances, and thus this algorithm needs to be modified.

One may argue how much one gains by working with intrinsic models instead of stationary ones, but there are many examples where the variograms seem to be unbounded over the region where observations are available. We think that in such situations one should not use a stationary model. In particular, estimation of both the cross-covariances and the pseudo cross-variogram requires the mean to be estimated first. This may lead to large biases when the variogran2, s are unbounded. One might further contest the need for modeling asymmetric cross-correlation patterns between variables observed in the two- or three-di-

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Generalized Cross-Covariances 781

mensional space because there is little evidence for this in the literature. But this lack is partly the result of the use of the cross-variogram for describing cross-correlation, and this function, being even, cannot describe asymmetries. For all these reasons, we believe that our method represents a significant ad- vantage.

G E N E R A L I Z E D CROSS-COVARIANCES

We consider a bivariate random process Z(x) = (Zl(x), Z2(x))' indexed by points x ~ ~2. (, denotes the transpose). The situation of more than two variables is a straightforward generalization. A linear combination

/11 /12

will be termed an increment (of order zero) if

nl /12

~Xj= ~

u ~ = O

i=

k=l

Thus, increments are unchanged if we add arbitrary constants to each variable. Note that the usual unbiasedness conditions in cokriging imply that the errors of the ordinary cokriging system are increments.

Definition (Z(x)) is intrinsic (of order zero) if (i) All increments have expectation zero.

(ii) The variance of any increment does not change if we add an arbitrary displacement h ~ y~2 to all x~, i and X2,k.

Because we consider here only increments and intrinsic models of order zero, we omit the order from our terminology.

We turn now to the computation of the variance of increments. Clearly War [in__~l

)kiZl(Xl,i) + k~=l

P'kZ2(X2,k )]

[ ~-~' ~iZl(Xl,i)l + Var [k~=l ltkZ2(X2,k)

]

= Var l_ i= 1

+ 2 Cov i )kiZl(Xl'i)' k~l= (1)

It is well known that

V a r X i Z l ( x , ~ = - E X i X / Y l l ( x l , i - x l j )

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782 Kiinsch, Papritz, and Bassi and

I

~ 1 n2 n2

V a r p, kZ2(X2,k) -~ -- ~ ~ 12kl, gl'Y22(X2, k - - X 2 , / ) ( 3 )

k l k = l l = 1

where %~(h) denotes the semivariogram of (Z~(x)). Because - 3 , ~ ( h ) acts like the usual autocovariance in (2) and (3), it also is termed a generalized autocovariance (see Matheron, 1973). Similarly a function p12(h) will be termed a

generalized cross-covariance

if

C o y _i= )kiZl(Xl,i), _{k 1}]/,kZ2(X2,k) = _{i =} _{k = l}~ hiPkPl2(Xl, i -- X2,k) (4)

holds provided ~ )k i : ~ /.t k = 0 (see Dowd, 1989 or Christakos, 1992, chapt. 4.4). However, it is far from obvious whether such a generalized cross-covariance exists for arbitrary intrinsic processes, whether it is unique and how we can determine it if it exists. Some simplification occurs because for an arbitrary x ~ ~2 nl n I ) k i Z l ( X l i) = =~1 )ki(Zl(Xl'i) -- Z I ( X ) ) i=l ' i= and Thus n2 //2

N u,k~(x2,~,) = k~l ~k(~(X2,k) - ~ ( x ) )

k=l C o y _{i =} )ki Z l ( x i, i), _{k = l} ~k Z2 (x2, k) /11 n2 = i~_l k~-_l ~'il-~kE[(Zl(Xl,i) - Z l ( X ) ) ( Z 2 ( X 2 , k ) - Z 2 ( x ) ) ]

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From this it follows that we have to check (4) only for special increments: PI2 is a generalized cross-covariance if

E[(Zl(x + hi) - Zl(x))(Z2(x + hE) -- Z2(x))]

= ~12(hl - h2) - r ' 1 2 ( h l ) - u12(-h2) + ~12(0) (6)

Equation (4) follows easily from (6) and (5) by using r~ )k i = y~ JLtk = 0. It would be nice if we could simplify (6) further and replace it by something which involves only a single displacement, but this seems not possible.

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We now discuss existence and uniqueness of generalized cross-covariances. First, note that if the process is stationary, then the usual cross-covariance is also a generalized cross-covariance. Next, for hi = h2 = h the left-hand side of (6) is twice the cross-variogram 712(h). Hence the generalized cross-covariance determines the cross-variogram

V12(h) = v12(0) - 89 + v 1 2 ( - h ) ) (7)

If the symmetry condition (iii) of Myers (1982) holds, then minus the cross- variogram is a generalized cross-covariance [see Myers, 1982, Eq. (16)]. To see the connection with Equation (7), note that the symmetry condition is equivalent to v12(h) = Vl2(-h) and constants do not matter for the generalized cross- covariance, as discussed after the main theorem. But if the symmetry condition does not hold, the cross-variogram does not determine the generalized cross- covariance. This is intuitively obvious from (7) and we will give such an example shortly. Next, assume that

1 Var[Zl(x + h) - Z2(x) ]

is independent of x for all h. Then, this expression is termed the pseudo cross- variogram "yq2(h) (Myers, 1991), and -TP2(h) is also a generalized cross-covariance [see Eq. (5') of Myers, 1991 or Eq. (6) of Papritz, Kfinsch, and Webster, 1993]. As discussed in Papritz, Kfinsch, and Webster (1993), the pseudo cross-variogram is shift-invariant for some, but not for all intrinsic nonstationary processes. It is easy to give examples for which neither the cross- variogram nor the pseudo cross-variogram nor the cross-covariances is adequate. Let (Y(l)(x)) and (Y(2)(x)) be two independent univariate processes with linear semivariogram g(h) = ][h[[ and define

ZI(X ) = allyr + a12yr

Z2(x ) = aElY~ - - U l ) -'F aE2y(2)(x - u2) Then, it is checked easily that

Pl2(h) : --alla211[U 1 -]- h{} - a128221lu2 + h[[

Thus, v12 =/: 712- Also, Cov[Zl(x + h), Z2(x)] and Var[Zl(x + h) - 7-,2(x)] both depend on x. Moreover, it is true that Var[XlZl(X + h) - X2Z2(x)] depends on x for any )x I and X2, so scaling of the variables does not help. Hence, the following result is a genuine extension to all situations considered before.

Theorem. If (Z(x)) is an intrinsic process with 711(h) and 722(h) both continuous at h = 0, then there exists a generalized cross-covariance v12. It is unique up to linear functions.

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784 K i i n s c h , P a p r i t z , a n d Bassi

into the Appendix. It shows that a valid generalized cmss-covariance can be represented in the form (27) where C12 and QI2 are measures satisfying (24) and (25), C i 2 is even and QI2 is odd. The first integral is equal to the cross- variogram, and Q12 is zero if the symmetry condition (iii) of Myers (1982) holds. In the stationary situation C12 and Ql2 are finite measures and we can integrate each term on the right-hand side of (27) separately. Then, we obtain the usual cross-covariance plus a linear function. It is easy to see that when a linear function c o + Clhl + c2h 2 with arbitrary coefficients Co, cl, c2 is added to a generalized cross-covariance, the result is again a generalized cross-covariance. This corresponds to the nonunJqueness of generalized covariances for univariate intrinsic processes of higher order (cf. Matheron, 1973).

By combining (1)-(3) and (4) we determine

+

1

nl nl n2 tt2 = - Z Z XA:rH(x,,, - x,,j) - Z /__E - i = l j = l k = l = nl n2 + 2 ~=1 Z _{i= k=l} )kiftkPl2(Xl, i -- X2,k) ( 8 )

This equation has two main uses. First, we can derive the equations for cokriging when 711, "Y22, and vl2 are given. This is well known, so we omit the details. Second, we can use it for estimating "/J2 as well as 711 and 722, although other methods are available for estimating the latter. This will be discussed in the next section. Thus, although the generalized cross-covariances are defined only through a rather abstract existence theorem, they can be estimated from data. So the concept also is useful for a practitioner.

E S T I M A T I O N O F G E N E R A L I Z E D C R O S S - C O V A R I A N C E S G e n e r a l Fitting via Least Squares

We assume that we have a parametric model vl2(h ) = Vl201; 0) and want to estimate the parameter 0 (which usually contains several components) from data

(Zl(xl,i),

i = 1 . . . . n 0 and (Z2(x2,i), i = 1 . . . . hE). Define

Pukl = ( Z l ( X l , i ) - - Z l ( X l , p ) ( Z 2 ( X 2 , k ) - Z2(x2,1)) (9) and

~ijkl(O) = PI2(XI,i - - X2,k; O) -- PI2(XI,i - - X2,1; O)

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Then by Equation (4)

E[Pijkt]

= 7rijkt(0) (I 1)

We now fit the Try/us to the cloud of the

Pijk~s

by least squares, that is we propose to estimate 0 by minimizing

nl nl n2 n2

QI2(0)

= ~ ~ ~ ~ (Pijkl-

71ijkl(O)) 2 (12)

i = l j = l k = l l = 1

[cf. Eq. (24) in Papritz, Kiinsch, and Webster, 1993]. There we assumed the symmetry condition which implies that vl2(h) = -3q2(h), but this is not necessary. The drawback of (12) is that in this form we need approximately

1 2 2 I 2 2

znl n2 multiplications and ~nl n2 additions to compute Q12(0) for one value of 0 even if we ignore the operations needed to compute 7ri/k~(0 ). Except for n 1 and n 2 being small, this is not feasible. However by simple algebraic manipulations we can convert the expression defining QI2(0) into a form which needs far less operations to compute. For this we write

so that

Then because

piA0) = vl2(xl.i - x2.~; 0)

"lFijkl(O) -~- P i k ( O ) - - D i l ( O ) - - D j k ( O ) q- P j l ( O )

7Fijkl ( 0 ) "~- - - 71"ijlk ( 0 ) "~- - - 71"jikl ( 0 ) ~" 7Fjilk ( 0 )

we obtain nl nl n2 n2 i j k=l t=l nl nl n2 n2

= 4 ~=l ~1 ~ ~=l

7rijkl(O)Pik(O) i = j = k = l 1= tl l n2 n I n2 n2

= 4nln2 i~1 ~=, Pik(0)2

--

4nl i~t k~=, t~l

Oik(O)Pil(O)

nl nl n2 Ill tll n2 n2

-4n2 ~=1 ~1 ~

_{i =} _{j =} Pik(O)Pjk(O) q-

4 ~=I ~J k~-Jl ~

Pik(O)Pjl(O)

k = l i j = l = 1 2

o

)

:

4n,n2 i~l

k~=l P i k ( 0 ) 2 - - 4n, i~l \ k : l P i k ( O ) 2 2 ( ~ _ _ ~ , ) 4 ( ~' "2 ) \

Pi~

- - 4n 2 ~ P i k ( O ) if" = ( 0 ) k = 1 i ~ i = I k ~ l

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786 Kiinsch, Papritz, and Bassi Similarly, because we have P i j ~ = - - P i~tk = - - e j i k l = P filk nl nl n2 n2 ~=1 X ~=1 PijklTrijkl(O) i = l j = k = l l = nl nl n2 /12

= 4 ~ ~=l ~ ~t PijklPik(O)

_{i = l j =} _{k=l l=} Ill n2 =

4nln2

i__~l

k_~l Dik(O)(Zl(Xl,i) -- Zl)(Z2(X2,k) - Z2) Taking these results together we see that

nl nl

88 QI2(O)

= / i - - ~ l j=~l ( Z l ( X l ' / ) -- Z I ( X I ' j ) ) 2 n2 n2 9 Z Z (Z~(x2,k) - Z2(x2.t)) ~ k=l I=I n l rt2 -- 2 n l n 2 i ~ l k=~l Pik(O)(Zl(Xl,i) -- 2 1 ) nl r/2 9 - § n , n 2

,x,

2 2

i=1 \ k = l 19ik(O) -- n2 k=l ~ i=l~

Pik(O)

2

+(i~=lk~=l#ik(O))

( 1 3 )

All sums on the fight now involve only two indices. This implies that the number of operations required to compute Q12(0) is only the square root of what we had before. So the computation is feasible for most datasets occurring in practice.

The simplest situation occurs when v~2(h; 0) depends linearly on the pa- rameters, that is:

P

v1201; 0) = ~ 0rfr(h ) (14)

r = l

wherefl(h) . . . fp(h) are given functions. Then Q12(0) is a quadratic function,

and minimizing

Ql2(0)

is equivalent to solving a system of linear equations.

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G e n e r a l i z e d Cross-Covariances 787

Experimental Generalized Cross-Covariances

In order to estimate generalized cross-covariances without imposing some particular structure, we use the model

P

~'12(h; 0) = ~ OrlArO! ) (15)

r = ]

where Al, A 2 . . . A r are subsets of B E and la(h) is the indicator function of a set A (i.e., equal to one if h ~ A and equal to zero otherwise). In the isotropic instance the sets Ar will be concentric rings around zero. In the anisotropic situation each ring will be split for instance into eight sectors.

We estimate the parameters 0r by the least-squares method described in the previous section. The normal equations are simple and involve only counting pairs of observation points satisfying certain conditions, but some thought is necessary for answering the question which parameters can be estimated. First, vl2 is determined only up to a linear function. So we can assume that Vie(0; 0) = 0, and VIE((1, 0)'; 0) = p12((0, 1)'; 0) = 0 in the anisotropic example. Second, some parameters cannot be estimated because of the geometry of the observation points: if xt, i - x2, k E A r for all pairs (i, k) or xl, i - x2, k ~ Ar for all pairs (i, k), then 0r cannot be estimated. In particular when two variables are never available at the same location, we cannot estimate a pure nugget com- ponent.

We consider the estimated parameter O r a s the experimental generalized

cross-covariance for lag h equal to the average of pairwise differences x~, i - xz, k belonging to Ar. For a more sophisticated procedure which gives an exper- imental generalized cross-covariance for all lags, one could replace the indicator functions in (15) by the basis functions in the space of splines with suitably selected knots, but we do not pursue this here.

Estimation in the Coregionalization Model

In the linear coregionalization model (Joumel and Huijbregts, 1978; Wack- ernagel, 1988; Goulard and Voltz, 1992), the symmetry condition is satisfied and the variograms have the form

S

"yjk(h; O) = ~ b}kg$(h, s )

s = l

Here, each g~ is a semivariogram of an intrinsic univariate process. The unknown parameters 0 are on one hand the coregionalization matrices (b]k) which have to be positive definite for all s and the parameters ~7 s which fix typically the range and/or the shape of each g~. In principle, we could apply our general estimation

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788 Kiinsch, Papritz, and Bassi method directly, but the nonlinearity in ~1 might be delicate, and the estimated coregionalization matrices are not guaranteed to be positive definite. Therefore, we prefer to build a procedure based on existing algorithms which avoid these problems. We use our new method only to compute an experimental cross- variogram, then we estimate iteratively the parameters ~s with the coregionalization matrices kept fixed and vice versa. In the first step, we use nonlinear least squares for the difference between experimental and theoretical variograms. For this step there exists a large experience and it generally behaves well. In the second step, we use the algorithm of Goulard and Voltz (1992) which iteratively minimizes the criterion

W S S = ~ , w ( h r ) t r [ ( V ( r (hr) - l'(hr; 0))) 2]

r

under the side condition that all (b}k) must be positive semidefinite. Here ~ is the matrix of experimental variograms, w are some weight functions and V is a diagonal matrix of weights. Compared to its standard version there are two particular features which needed attention. First, as explained, if the sampling locations of the two variables are disjoint and gi corresponds to a nugget effect then the experimental cross-variogram is determined only up to the addition of const.gl(h) where const, is arbitrary. Thus, it is not possible to estimate blz. We tackle this problem as follows: At the beginning of the ith iteration step we replace '~i2 by ~12 - const.(~ I where

= Z W r [ ' Y l 2 ( h r ) -

-Yi2(hr; o'i-I')]/Z

W r const. (0

r I r

and 0 (;" 1) are the parameters estimated in the (i - 1)th step. During the ith iteration we then minimize WSS as usual with respect to bil, bl2, and bl2 subject to the condition that the first two coefficients should be nonnegative. The coef- ficient b12 tends to zero with proceeding iteration and therefore does not interfere in estimating bll and bl2. Second, in the undersampled situation, the experimental variograms "Y11, "YI2, and "~22 are all available at different lags. But for the criterion WSS, we need them for the same lags. We solve this by setting the missing values of the experimental auto- and cross-variograms equal to 71/(h; 0 (i- ~)) at the begin of the ith iteration step. A FORTRAN program which puts all these steps together is also available upon request from the second author.

NUMERICAL EXPERIMENTS

The Experimental Cross-Variogram

We consider here the situation where we have observed a bivariate Gaussian random process at coincident locations on the regular grid {1, 2 . . . n} 2. In

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Generalized Cross-Covariances 789

this situation, three experimental cross-variograms are available: the standard one which averages the products

(Zl(xi) - Z l ( x j ) ) ( Z A x , ) - Z2(xi))/2

over pairs (xi, xj) in a given lag class, the new one introduced in this section and the one based on the empirical cross-covariances t~2(h), which we obtain by averaging

(ZI(Xi) -- Z 1 ) ( Z 2 ( x j ) - Zr2)

over pairs (xi, xj) in a given lag class. Our aim is to compare these three estimators with respect to bias and standard deviation. They are all certain quadratic forms in the observations. W e thus can compute their expectation and because o f the Gaussian distribution also their variance in closed form. W e first look at the situation where all auto- and cross-variograms are linear:

~ff(h) = ~011hll

We select fl~ = ~22 = 1 and/3~2 = 0.7 although this is not essential for our conclusions [note that the bias is proportional to fl~2 and the standard deviation is proportional to (/311fl22 + fl~2)~/2]. The results for n = 4, 8, 12, 16 and lag classes (i - 0.5, i + 0.5] are given in Tables 1-3. Clearly the estimator based on experimental cross-covariances is the worst both with respect to bias and random variability. O f course this is not surprising because our model is not stationary. But it supports our claim that one should not rely on experimental cross-covariances when variograms are unbounded. A similar behavior is expected also for the experimental pseudo cross-variogram. In contrast to this our new estimator has negligible bias and shows the same random variability as the

lble 1. Standard Deviation for the Standard Experimental Cross-Variogram. This Estimator is Unbiased for All Lag Classes

Lag class 1 2 3 4 5 6 7 8 9 10 11 = 4 0.43 1.12 1.95 2.81 . . . . = 8 0.23 0.62 1.14 1.91 2.85 3.73 4.64 5.62 6.82 8.54 - - = 12 0.16 0.44 0.81 1.38 2.09 2.83 3.67 4.52 5.49 6.47 7.39 = 16 0.13 0.34 0.64 1.09 1.65 2.25 2.94 3.68 4.56 5.50 6.38 Lag class 12 13 14 15 16 17 18 19 20 21 = 12 8.40 9.61 10.85 12.27 13.43 . . . . . = 16 7.28 8.21 9.17 10.04 11.18 12.32 13.86 14.89 16.18 17.33

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7 9 0 K i i n s c h , P a p r i t z , a n d B a s s i E 0 >. c- O e~ e~ f~ e~ ' , 0 c-,t f-4 I ~ t t 3 o 0 I } I I I t"4 II II II II tt3 t ~ tt~ O II II

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G e n e r a l i z e d C r o s s - C o v a r i a n c e s 7 9 1 e - E > e ~ r ~ A~ I I I I I I II II I[ U p ~ o O II II

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792 Kiinsch, Papritz, and Bassi standard estimator except lor the lag classes which are small compared to the observation region. For these lags the standard estimator is considerably better. We explain this behavior as follows: our new estimator has to estimate the cross- variogram simultaneously for all lag classes, and the large variability for larger lags also affects the smaller lags. An improvement can be obtained from the following idea: instead of computing the estimator based on all observation locations, we can compute it for smaller, overlapping subsets and then average over all the results. By doing this we obtain estimates only for small lag classes, but it is for these that we want an improvement. We implemented this idea by selecting as the subsets all two by two squares of neighboring locations, then we have only one lag class, the first one. The standard deviation of the resulting estimator is 0.42 for n = 4, 0.22 for n = 8, 0.15 for n = 12, and 0.12 for n = 16, respectively. Comparison with Table 1 shows that this estimator is as good as the standard one.

A further estimator is given by

~12(h) = "~P2(O) - .~P2(h)

where -~Px(h) denotes the experimental pseudo cross-variogram which is obtained by averaging

(EI(Xi) -- Z l -- Z2(xj) "q- Z2) 2/2

over pairs (xi, xj) in the lag class corresponding to h. A comparison of this with the previous estimators is more difficult because the conclusions depend on the selection of ~ij. For instance the bias is not proportional to ~ 2 , but proportional to/311 + /~22 - 2/312 and thus is large if the two processes are independent or negatively dependent. In the special situation ~1 = ~ 2 2 = 1 and /312 = 0.7

which is favorable for this estimator, the biases are approximately half as big as for the estimator based on empirical covariances and the standard deviations are similar to the ones for the standard experimental cross-variogram.

We did the same calculations for exponential variograms 7u(h) = /3ij(1 - exp(-l[hf]/2))

This model is stationary, and we thus expect that all three experimental cross- variograms are comparable. The results show that this is indeed the situation; both the biases and the standard deviations of all three estimators are close. We thus omit the corresponding tables.

Simulation Results for Some Coregionalization Models

We consider here all possible nine combinations of three models and three sampling configurations. In the first model

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with 311 = 322 = 1,312 = 0.7, ~ = 2, in the second model

']/jk(h) = ~)kX[[[hl] >0] 4- ~2k(1 -- exp(-Ilhll/w)) (17) with 311, = 3~2 = 0.25, 3112 = 0.175, 3~, = /3222 = 0.75, /322 = 0.525, and r/

= 2. Finally, in the third model

3`jkoa) = Ilhll" (18)

with fill = fl2z = 1, fl~2 = 0.7, r/ = 1. The unknown parameters are all the fls and r/.

In the first configuration we have n 1 = n2 = 256 with

Xl, i = X2, i ~ {1, 2 . . . 16} 2 (19)

in the second configuration again n~ = n2 = 256, but

Xl,i e {1, 2 . . . 16} z, x2, i E { 1 . 0 0 1 , 2 . 0 0 1 . . . 1 6 . 0 0 1 } 2 (20) and in the third configuration n~ = 400, n 2 = 100

X,,i ~ {1, 2 . . . 20} z, X2,i ~ {1, 3 . . . 19} 2 (21) These options cover a wide range o f different situations.

For each model and configuration we generated 200 replicates o f (Zl(xl.i)) and (Z2(x2,i)). The standard estimator was used to compute all the experimental autovariograms, and it also provided estimates of the experimental cross-variograms for configurations (19) and (21). For all configurations the experimental cross-variograms were computed by the new estimator using the procedure as outlined. W e used the lag classes (0, 0.5], (0.5, 1.5] . . . . for all variograms, then we fitted the linear model of coregionalization to the experimental variograms applying Goulard and Voltz's algorithm with the extensions as described. Experimental semivariances for lag distances <_ 8 were used in fitting, and all the lag classes had equal weight.

The biases and standard deviations of the estimated parameters were computed from the 200 replicates. They are listed in Tables 4 - 6 . T w o types o f comparisons are of interest here: comparing the two estimators for the same configurations and comparing the different configurations for the same estimators. For configuration (19) and models (16) and (17) the two estimators are practically the same. For configuration (21) and models (16) and (17) the new estimator is clearly better. This is true not only for the parameters o f ~21 but also for those o f 3'22.

Surprisingly, the new estimator of the range r/is better only in model (16), but not in model (17). For (17) a smaller bias of the new estimator ~ is com- pensated by a larger standard deviation. With the model (18) the standard estimator is better for both configurations (19) and (21) as was expected. The

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7 9 4 K i i n s c h , P a p r i t z , a n d B a s s i

T a b l e 4. Biases (Upper) and Standard Deviations (Lower) of the Parameters o f the L i n e a r C o r e g i o n a l i z a t i o n M o d e l Fitted to Experimental V a r i o g r a m s for Configuration (19) ~

M o d e l (16) M o d e l (17) M o d e l (18) O L D N E W O L D N E W O L D N E W 17 0.065 0.064 0.494 0 . 5 0 6 - 0 . 0 7 5 - 0 . 0 7 6 0 . 7 8 4 0.781 2.008 2 . 0 7 7 0 . 2 6 2 0 . 2 8 5 131~ 0.034 0.033 - 0 . 0 1 3 - 0 . 0 1 2 0 . 0 8 6 0.095 0.295 0.293 0.151 0.151 0.303 0 . 3 3 4 /312 0.032 0.032 - 0 . 0 0 6 - 0 . 0 0 5 0 . 1 3 6 0.143 0.283 0 . 2 8 6 0.145 0.143 0.315 0.342 /~1t2 0 . 0 2 9 0.029 - 0 . 0 1 2 - 0 , 0 1 4 0.058 0 . 8 4 0.237 0.218 0.111 0 . 1 1 4 0 . 1 7 2 0 . 2 2 0 B~I - - - - 0,059 0,061 - - - - - - - - 0.319 0,333 - - - - fl~2 - - - - 0.058 0.059 - - - - - - - - 0.312 0 . 3 2 0 - - - - /~22 - - - - 0.057 0.062 - - - - - - - - 0.291 0.298 - - - -

~ experimental cross-variograms w e r e computed b y the standard ( O L D ) and new ( N E W ) esti-

mators.

T a b l e 5, Biases (Upper) and Standard Deviations (Lower) o f the Parameters o f the L i n e a r Coregionalization Mode] Fitted to E x p e r i m e n t a l V a r i o g r a m s for Configuration (20) ~

M o d e l M o d e l M o d e l (16) (17) (18) 7/ 0.077 0 . 2 5 0 - 0 . 0 9 8 0 . 7 3 6 1-376 0 . 2 4 4 B~ 0.024 - 0 . 0 1 2 0 . 0 8 0 0.266 0-123 0.299 /3212 0.002 - 0 _ 0 1 4 0.124 0.248 0.121 0.333 BI2 0.017 - - 0 . 0 3 0 0.216 - - 0 . 2 2 0 /~1 - - 0 . 0 7 2 - - - - 0.303 - - t ~ - 0 . 0 5 8 - - - 0 _ 2 8 2 - - r - 0 . 0 5 0 - - - 0.241 - -

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G e n e r a l i z e d C r o s s - C o v a r i a n c e s 7 9 5 > e- - = o = o v 0~ = o e~ A ~ O c ' 4 0 ce~ ~ c~ O C'4 I I I I U-- ~ ~ I ~ I ~ ~ ~ ~ ~ " ~ ~ ~ U "~ ~ W'~ I I I I O ~ O O O O O O O O O O O O I O ~ O O O O O ~ O O O O O O I O O O O O O O O O O O O O O O O o z f~ e~ e ~ E 8 O - = ,> e~ - = 2

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796 Kiinsch, Papritz, and Bassi larger standard deviation of the new estimator of the experimental cross-variogram at small lags carries over to the estimator of the parameters.

Comparing the different configurations, we see that (21) is the best for model (16). The only exception here is the bias of the estimated range. For model (17) configuration (19) is the worst, but the comparison between (20) and (21) is less clear. Using (20) we obtain a greater precision for the estimates of

r/,/3212,

and/322, but we cannot

estimate

ill2 at all and we loose some precision for/3~1. Finally for model (18), configuration (19) again is worst, whereas (20) and (21) are about the same. The precisions for ~ and fill are practically the same for both configurations, and (20) is better for/312, but worse for/322. As a conclusion, taking an equal number of observations for the two variables at coincident points seems to be a bad strategy. Surprisingly undersampling of one variable seems to allow reasonable estimates even for the autovariogram of the undersampled variable. Of course for selecting between different configurations, sampling costs also have to be taken into account.

C O N C L U S I O N S

The generalized cross-covariances solve some of the difficulties associated with cross-variograms, pseudo cross-variograms, and cross-covariances. The last two concepts restrict the class of models that we can consider. Our numerical experiments have clearly shown that experimental cross-covariances should be avoided when variograms are unbounded over the available lags. Unlike the cross-variogram the generalized cross-covariances can be used for modeling correlation between intrinsic random processes that depend on direction. The new method for estimating generalized cross-covariances allows us to check this, and it solves the problem of computing the experimental cross-variograms when some variables are undersampled with respect to others.

We have implemented the new estimator, and for the linear model of coregionalization where the generalized cross-covariances are equal to minus the cross-variogram, we have combined it with the algorithm of Goulard and Voltz (1992) to ensure a valid model. Tests on simulated data showed that the new estimator of the cross-variogram generally behaves well. If, however, the variograms are unbounded then it is less efficient at short lags than the standard method. But the latter requires coincident locations, and we have indicated how the former can be improved in such a situation.

In summary, we believe that our results solve some problems related so far with cokriging and enhance this powerful tool.

R E F E R E N C E S

Chfistakos, G., 1992, Random field models in earth sciences: Academic Press, San Diego, Cali- fornia, 474 p.

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Generalized Cross-Covarianees 797 Dowd, P. A., 1989, Generalised emss-covariances, in Armstrong, M., ed.; Geostatistics, v. 1:

Kluwer Acad. Pub. Dordrecht, The Netherlands, p. 151-162.

Gelfand, I. M., and Vilenkin, N. Ya., 1964, Generalized functions, Vol. 4, Applications of har- monic analysis: Academic Press, New York, 384 p.

Goulard, M., and Voltz, M., 1992, Linear coregionalizatlon model: tools for eslimation and choice of cross-variogram matrix: Math. Geology, v. 24, no. 3, p. 269-286.

Joumel, A. G., and Huijbregts, C I., 1978, Mining geostatistics: Academic Press, London, 600 p.

Matheron, G., 1973, The intrinsic random functions and their applications: Adv. Appl. Probab. v. 5, p. 439-468.

Myers, D. E., 1982, Matrix formulation of co-kriging: Math. Geology, v. 14, no. 3, p. 249-257. Myers, D. E., 1991, Pseudo-cross variograms, positive-definiteness, and cokriging: Math. Geology,

v. 23, no. 6, p. 805-816.

Papritz, A., Ktinsch, H. R., and Webster, R., 1993, On the pseudo cross-variogram: Math. Ge- ology, v. 25, no. 8, p. 1015-1026.

Wackernagel, H., 1988, Geostatistical techniques for interpreting multivariate spatial information,

in Chung, C. F., Fabbri, A. G , and Sinding-Larsen, R., eds., Quantitative analysis of mineral and energy resources: D. Reidel Publ. Co., Dordrecht, The Netherlands, p. 393-409. Wackernagel, H., 1995, Multivariate geostatistics: an introduction with applications: Springer,

Berlin, 256 p.

APPENDIX: THE EXISTENCE OF GENERALIZED CROSS- COVARIANCES

We give two proofs. The first one is short, but relies on abstract analysis. According to Gelfand and Vilkenkin (1964, chapt. III 5.5), there exists a complex matrix valued measure (F0) 1 _< i,j ~ z on N2 with the properties

(dF 0

(to)) is positive definite (22)

dFo(-to ) = dF~(to)

(23) I Iltoll2dFo(oJ) exists (24) 0< lltd-< 1

f

dFij(to)

exists (25) II~[I > 1

f

E

= / X k exp ( x / - l t o 9 x~) d tt~II > o k

9 ~,, X; exp (_~-Z-~to .

x;)dFo(to)

z (26)

for any two increments Ek X~Zi(xk), Et X~ Zj(x;) where the dot denotes the inner product in ~ 2

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798 Kfinsch, Papritz, and Bassi

Let us c o m m e n t briefly on the differences between this and the statements in Gelfand and Vilenkin (1964, chapt. III 5.5). Condition (25) is satisfied because we consider only usual random fields (Z(x)) and not generalized fields (Z(~b)). The relation (23) holds because our random fields are real and not complex. Conditions (22) and (24) are the same as in Gelfand and Vilenkin (1964). Finally (26) is Equation (17) in Gelfand and Vilenkin (1964, p. 360) for ~b(x) = ~ kkS~(x) and ~b(x) = r. k[Sx~(x). The term E

aijpqOtp-~q

is missing in (26) because we assumed that all increments have expectation equal to zero. Using (26) it is not difficult to obtain an explicit expression for the generalized cross-covariance. Write dF12(to) =

dC12(to) + xf-S-ldQ12(to)

(decomposition into real and imaginary part). Then, by (23)

dCl2

is even and

dQ12

is odd. Thus

= I ~ kkk[ cos (to 9 (xk - x~))dC12(to) d lkoll > o

k,1

-- t ~ Xkkl sin (to - (Xk -- x[))dQl2(to) 0 Iloll > o k,t

Note that both integrals exist because o f r~ k Xk = ~ l )k[ : 0 and (24). But we cannot exchange directly the integral and the sum because the integrals would then no longer exist. H o w e v e r

X k x ; c o s ( t o 9 (xk - x ; ) ) = E X k X ; [ c o s ( t o 9 (x~ - x ; ) ) - 1]

k,l k,I

because r. Xk = r. hi = 0, and cos(to 9 h) - 1 is integrable by (24) and (25). Similarly

Z kkk/sin(to " (Xk -- xD)

k,l

= ~ )~kk[

[sin(to 9 (Xk -- X/)) -- to " (Xk -- x/)l[Ikd_< Ill

k,1

and sin(to 9 h) - to 9 hltll,oll < 1] is integrable because it is bounded and behaves as - ( t o 9 h)3/6 for Iltoll --} 0. Hence a generalized cross-covariance is given by

vl2(h) = llltoll >o

- I[koll > o

I

and the first p r o o f is completed.

[cos(to 9 h) - 1]dC12(to)

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The second proof is by direct construction. H o w e v e r checking the necessary details is long and delicate, so we only sketch the arguments. T h e idea is to use (6) for sufficiently many his and h2s in order to obtain v12.

W e first decompose v12 into an even and an odd part: v~-2(h) = (v12(h) + v l 2 ( - h ) ) / 2 v~2(h) = (vl2(h) - v l 2 ( - h ) ) / 2

so that v12(h) = v~(h) + v~(h). Then using (6) for hi = h2 = h and putting v~2(0) = 0 without loss of generality we determine

v~-2(h) = 89 + h) - Zl(x))(Z2(x + h) - Z2(x))]

So v~- 2 is simply the cross-variogram. Next selecting hi = h, h2 = h/2 in (6) we obtain

' ,,~(h)

v~(h/2)

= 8 9 +

+ 89 + h) - Z~(x))(Z2(x + h/2) - Z2(x))l (28)

Without loss of generality we may take v{-2(e~) = v~-2(e2) = 0 where el = (1, 0) and e2 = (0, 1) (v12 is determined only up to a linear function). So (28) gives v~-2(2-nei) for n = 1, 2 . . . But by (6) we obtain v12(h I + h2) from Vl2(hl)

and v12(-h2). So by repeated application of (6) we determine

v12(k12 -n', k22-n2), that is we have constructed vl2 for all lags with dyadic coordinates. Note that we can arrive at the same lag (kl2 -n~, k22 -n2) by many different sequences o f intermediate steps. So for a rigorous p r o o f we would have to show that the result does not depend on this. Finally, in order to go from dyadic to general lags we have to show that v12 restricted to dyadic lags is continuous. Essentially this can be done by using Schwarz' inequality and the continuity of 3'~ and 3'22 repeatedly.